For the sixth day of our advent countdown to Newton’s birthday, we have the first equation that really departs from the usual notation. I’ve gotten to kind of like the way the Matter and Interactions curriculum handles this, though, so we’ll use their notation:

This is what Chabay and Sherwood refer to as the Energy Principle, which is one of the three central principles of mechanics. The term on the left, ΔE represents the change in the total energy of a system, while the two terms on the right represent the work done on that system by its surroundings, and any heat energy flow into or out of the system due to a difference in temperature.

So, why is this important?

This is an equation that is subject to the Indiana Jones and the Last Crusade rule: If you choose wisely, things will go well. If you choose…. poorly, well…

Any time you’re dealing with a bunch of multiple interacting objects, it’s usually possible to choose to define the “system” such that the two terms on the right are zero. In which case, this equation becomes the more familiar statement of the conservation of energy: The total energy of the system at the start of the problem is equal to the total energy of the system at the end of the problem. A wise choice of system thus turns physics into accounting: you have a fixed amount of energy, and can change how you apportion that energy between the subsystems. It’s usually easy to calculate the energy of some of the subsystems, which makes finding the energy of the others a simple matter of arithmetic.

It’s worth writing this out in its full form, though, to emphasize that the total energy of an arbitrarily chosen system is not necessarily zero. If objects outside your system of interest are able to interact with it in ways that do work or add heat, then the total energy can change, and life becomes a little more complicated. That’s an important fact to remember for any case in which your system is smaller than the whole universe.

But does this have any deeper lesson for us? Well, the energy principle is ultimately a result of Emmy Noether’s famous theorem relating symmetry to conservation laws. The underlying reason why the total energy of a closed system (that is, one not interacting significantly with outside objects) is conserved is that the laws of physics are symmetric in time: they will work the same way tomorrow that they did yesterday. According to Noether’s theorem, this means that there must be some quantity that remains constant as you move forward in time, and that quantity is the energy.

(Noether’s theorem also explains momentum conservation: the laws of physics are symmetric in space: they work the same way in Brisbane, Australia as in Niskayuna, New York. According to Noether’s system, this means there must be some quantity that remains constant as you move from one place to another, and that quantity is the momentum.)

So, conservation of energy is much more than turning out the lights when you’re not home. It’s built into the deep structure of the universe, and one of the most fundamental properties of physics.

Come back tomorrow for the next great equation of the season, as we continue counting downt he days to Newton’s birthday.

Comments

I recall once reading about somebody (online, of course) arguing against evolution through a conservation-of-energy/entropy trends to a maximum argument: if evolution were true, then there would be increased order/complexity as organisms developed, which violated the entropy principle.

As the person relating the story to me put it, this particular crank apparently forgot about the big energy-generator 1 AU from the Earth, putting additional energy into the system…

which superficially look like the first and second laws of thermodynamics, though they’re not. They are:

1. “Nothing can come from nothing” and
2. “No cause can produce or give rise to perfections or excellences that it does not itself possess”.

The first is the basis for the cosmological argument for the existence of God, and the second is the basis for believing that evolution can’t happen without divine guidance. If you squint at them, the first one looks sort of like the conservation of mass/energy, and the second looks sort of like the law that entropy always increases in a closed system, if you read entropy as disorder or imperfection. So you can translate old Christian apologetics easily into pseudoscientific language by simple substitution.

Of course, once you try to be precise about terms and take all the real physical caveats into account, the translation breaks down. There’s no real reason to believe that conservation of mass/energy applies at the moment of the Big Bang, and it even has to be severely modified in general-relativistic cosmology, especially in the presence of dark energy or inflation; and these are precisely the situations that are relevant to the cosmological argument. And the creationist use of the second law is even worse, since entropy is not really the same thing as lack-of-excellence, and the biosphere is nothing like a closed system.

The law in this post is the 1st law, which doesn’t specify a direction that energy has to flow in. (∆E could be getting bigger or smaller, and Q and W could be positive or negative also.) It’s the 2nd law that says heat has to flow from higher temperature to lower, or not flow at all.

As for the argument that life can’t exist if the entropy of the whole universe is always increasing, that goes back to where you put the boundaries of your system. If your system is everything inside an imaginary surface that encloses a human body, for instance, the entropy of everything in the system can DECREASE, so long as the entropy of the surroundings INCREASE by the same or a greater amount. This happens through heat transfer through the boundary of the system. When you add up the total entropy of the system plus the surroundings, it’s always bigger, but the system can decrease in entropy.

Books

You've read the blog, now try the books:

Eureka: Discovering Your Inner Scientist will be published in December 2014 by Basic Books. "This fun, diverse, and accessible look at how science works will convert even the biggest science phobe." --Publishers Weekly (starred review) "In writing that is welcoming but not overly bouncy, persuasive in a careful way but also enticing, Orzel reveals the “process of looking at the world, figuring out how things work, testing that knowledge, and sharing it with others.”...With an easy hand, Orzel ties together card games with communicating in the laboratory; playing sports and learning how to test and refine; the details of some hard science—Rutherford’s gold foil, Cavendish’s lamps and magnets—and entertaining stories that disclose the process that leads from observation to colorful narrative." --Kirkus ReviewsGoogle+

How to Teach Relativity to Your Dog is published by Basic Books. "“Unlike quantum physics, which remains bizarre even to experts, much of relativity makes sense. Thus, Einstein’s special relativity merely states that the laws of physics and the speed of light are identical for all observers in smooth motion. This sounds trivial but leads to weird if delightfully comprehensible phenomena, provided someone like Orzel delivers a clear explanation of why.” --Kirkus Reviews "Bravo to both man and dog." The New York Times.

How to Teach Physics to Your Dog is published by Scribner. "It's hard to imagine a better way for the mathematically and scientifically challenged, in particular, to grasp basic quantum physics." -- Booklist "Chad Orzel's How to Teach Physics to Your Dog is an absolutely delightful book on many axes: first, its subject matter, quantum physics, is arguably the most mind-bending scientific subject we have; second, the device of the book -- a quantum physicist, Orzel, explains quantum physics to Emmy, his cheeky German shepherd -- is a hoot, and has the singular advantage of making the mind-bending a little less traumatic when the going gets tough (quantum physics has a certain irreducible complexity that precludes an easy understanding of its implications); finally, third, it is extremely well-written, combining a scientist's rigor and accuracy with a natural raconteur's storytelling skill." -- BoingBoing